Lemma for psgnuni26833. Given two consequtive transpositions in a
representation of a permutation, either they are equal and therefore
equivalent to the identity, or they are not and it is possible to
commute them such that a chosen point in the left transposition is
preserved in the right. By repeating this process, a point can be
removed from a representation of the identity. (Contributed by Stefan
O'Rear, 22-Aug-2015.)

Lemma for psgnuni26833. It is impossible to shift a transposition
off the
end because if the active transposition is at the right end, it is the
only transposition moving in contradiction to this being a
representation of the identity. (Contributed by Stefan O'Rear,
25-Aug-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)

Lemma for psgnuni26833. An odd-length representation of the
identity is
impossible, as it could be repeatedly shortened to a length of 1, but a
length 1 permutation must be a transposition. (Contributed by Stefan
O'Rear, 25-Aug-2015.)

If the same permutation can be written in more than one way as a product
of transpositions, the parity of those products must agree; otherwise
the product of one with the inverse of the other would be an odd
representation of the identity. (Contributed by Stefan O'Rear,
27-Aug-2015.)

The operator which multiplies an MxN matrix with an NxP matrix. Note
that it is not generally possible to recover the dimensions from the
matrix, since all Nx0 and all 0xN matrices are represented by the empty
set. (Contributed by Stefan O'Rear, 4-Sep-2015.)

Define the adjunct (matrix of cofactors) of a square matrix. This
definition gives the standard cofactors, however the internal minors are
not the standard minors. (Contributed by Stefan O'Rear, 7-Sep-2015.)